Annals of Mathematical Logic 11 (1977) 105-145 © North-Holland Publishing Company

ADDING DEPENDENT CHOICE

David PINCUS* Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A.

Received 10 June 1975

Let ZF be Zermelo-Fraenkel set theory and DC be the principle of dependent choice. This · paper presents a uniform method for transforming a ZF model of a statement <P into a ZF + DC model of C/>.

0. Introduction

Adding dependent choice (DC) to the consistency of the statement cf> refers to the process of transforming a Zermelo-Fraenkel (ZF) model of cf> into a ZF + DC model of ef>. A similar definition could be given for adding some other statement, x, to. the consistency of ef>. Adding DC to the independence of x from cf> is the same as adding DC to the consistency of cf> 11 - X· For the reader's convenience a glossary at the end of this paper contains the complete statements and abbreviations for the various choice principles discussed here.

There is no completely general method for adding DC since DC itself is independent of ZF (i.e . from a tautology). On the other hand, many independences have nothing to do with DC but all their known models fail to satisfy DC. A method of adding DC to such independences is philosophically interesting since the classical mathematics of the real numbers can be carried out in ZF + DC. ZF alone is insufficient for classical mathematics as is illustrated by a model of Levy [12] in which the continuum is a countable union of countable sets.

The basic" method for adding DC is illustrated in Section 1 with the thematic independence of the axiom of choice (AC) from the ordering principle (0). A variant of this model is previewed in [19]. The important lemmas, sketched there, are proved in full here. I am unable to make good my claim of [18] and add DC to the independence of AC from the prime ideal theorem (PI) and PI from the Hahn Banach theorem (HB). Section 1 concludes with a discussion of this problem and my partial results. R. Solovay and I have recently succeeded in adding DC to the independence of PI from HB. We use an entirely different method.

Section 2 illustrates how to add DC to a number of other independences. The independence of 0 from the axiom of choice for families of finite sets, C<w, is considered in detail. The resulting model satisfies the countable multiple choice

• Supported ,in part by NSF grant GP 44014.

105

106 D. Pincus

axiom Z(w ). Hence the independence of 0 from Z(w) is proved, answering a

question of Sageev [22] . Section 3 shows how to add Dc<•, K a regular cardinal , without adding sets of

cardinal < K . This turns out to involve a new idea, that of a hereditarily almost

disjoint (HAD) function. A question of Levy [12] is also answered by showing the

independence of c. (choice for families of K -size sets) from c <. for regular K. I do not know how to do both results at once i.e . to add Dc<• to the independence of c.

from c <•· The last two sections contain more applications of HAD functions. Section 4

extends my results [17] on the ZF transfer of Fraenkel-Mostowski independences. A class of transferable statements is given which includes, in the terminology of

[17], the class 2 statements and the class I statements in which the instances of C are bounded. Section 5 shows how to add countable choice, cw, without adding DC.

The thematic independence of AC from 0 is considered in detail. A number of quest ion s arise in the course of the paper. The three which most

interest me are the following: (!) Can DC be added to the independence of AC from PI?

(2) Can DC be added to the independence of Cw, from Cw? (3) Can c <On (choice for well-ordered families) be added to any of the indepen­

dences discussed here, in particular to the thematic one of AC from 0?

1. The basic method

Theorem I. DC can be added to the independence of AC from 0.

The ZF independence of AC from 0 was established by Halpern and Levy ([1]

and [8]) using one of Cohen's original models for - A C. The model is U[ Io] where

U is a model of ZFE (E is Godel's axiom of strong choice) and Io is an independent

set of Cohen reals . DC fails in this model since Io is a Dedekind finite set relative to

U[Io]. How does one remove the failure of DC? The simplest approach is to add,

relative to U[Iu], a generic map from w into l o. It is easy to see that this adds DC to the model , in fact it adds AC. Suppose one instead adds an independent set / 1 of

generic maps from w into Io. AC is no longer true because !1 now becomes Dedekind finite but Io has become countable. The contradiction to DC has been

pushed one step farther out, from Io to /1. The basic idea for adding DC is to iterate the "pushing out" process through w 1•

Thus one adds to U a sequence (I,. : a< w1). I a+ l is an independent set of

enumerations of I". For a limit I" is an independent set of choice functions for the sequence (Jfl : f3 < a). This will completely eradicate counterexamples to DC but

AC will still be false since (I" : a < w1) will fail to have a choice function in U(J" : a < w1). One now shows that the proof of 0 in U[Io] is sufficiently uniform to

Adding dependent choice 107

generalize to U(Ia :a < w,). This is possible but difficult. I have been unable to show the corresponding fact about PI , which also holds in U[Io] (see ([8]).

The model of [19], Section 3 is like the one described above except that the

iteration is carried through w instead of w, . This section duplicates some elemen­

tary arguments of [19]. It also proves in detail the lemmas which were only sketched

there. (See Lemma 1.5 and 1.8(c).). The reader familiar with [19] will have no trouble proving that Z(w) and the existence of a class of cardinal representatives

hold as well in the model of this section. The presentation of forcing in this paper is basically that of Cohen [ 1], which is

particularly suited to the treatment of inner models of generic models. The idea of

iterated forcing as in [23] underlies the above heuristic discussion of the model. It

does not seem necessary or even desirable in the formal presentation.

1.1. Elementary facts and notation

(a) Finite sequences If s, t, and u are finite sequences, st denotes the concatenation of s and t, s ~ t

means that s is an initial segment of t, Sub (s, t, u) denotes the sequence tv just in

case u = sv. Obvious properties of Sub will be used implicitly throughout e.g .

u =Sub (t, s, Sub (s, t, u )).

(b) Equivalence relations Let R and S be binary relations on X and Y respectively. R f Z = R n (Z x Z).

R and S agree if R f X n Y = S f X n Y. TC (R, S) denotes the transitive closure

of R and S defined on XU Y. TC(R, S, T) = TC(TC(R, S), T) = TC(R, TC(S, T)) etc. If R and S are equivalence relations so 'is TC(R , S). If both are equivalence

relations and R f X n Y C Sf X n Y then TC(R , S) agrees with R. If R and S agree and both are equivalence relations then TC(R, S) agrees with both R and S. Furthermore, in the last situation if x EX andy E Y then x bears TC(R, S) toy if

and only if for some z E X n Y , sRzSy. This generalizes to infinity many relations

R. if all the X.. n X 13 are the same for a -1 {3.

(c) Functions Iff is a function Domain f and Range f have their usual meanings. Parentheses

will often be omitted in the denotation of function values i.e. fx will be written

instead of f(x ). If f is a function valued function fxy stands for (fx )y etc.

1.2. Constants

In Cohen's treatment [I] of forcing the extending model is built from 3 sorts of

constants: ground model constants, undefined (new) constants, and abstraction

constants. These constants are arranged in a hierarchy which is completely

determined once one has assigned levels to the new constants. In the resulting

model each element is named by several constants. To avoid excessive notation the

108 D. Pincus

same symbol is used for the model element and the constant which names it. In the

occasional ambiguous situation x denotes a constant which names x. The present model is built from w1 new constants called new function symbols

and a single other new constant, I. The new function symbols are assigned orders

a < w 1• w many new function symbols are assigned to each order. The new function

symbols of order 0 are inserted in the hierarchy at level w. The new function

symbols of order a are inserted in the hierarchy at the first level above the one

containing all abstraction constants of the form ({3, f) where {3 <a and f is a new

function symbol of lower order. I is inserted in the hierarchy at the first level above that of all new function symbols.

1.3. Terms

Terms are nonempty finite sequences which are used in conditions to keep track

of induced equalities and inequalities between values of new functions. More

motivation for their use appears in [19]. Terms are assigned orders a E w 1 U {- 1} .

(a) The te rms of length I are exactly the new function symbols, 0 , and I. A new

function symbol has its assigned order. 0 and I have order - 1.

(b) If t is a term of order a + I , a E w1 U {- I} then the !-element extensions of t which are terms are exactly those sequences of the form tn, nEw. tn has order a. (c) If t is a term of limit order a then the !-element extensions of t which are terms

are exactly those sequences of the form 1{3, {3 <a. t{3 has order {3.

(d) If t has order - I then no extension oft is a term. Notice that if tis a term and

s ~ t then s is a term . Also if s, t, and u are terms, s ~ u, and s and t have the same

order then Sub (s, t, u) is a term . From now on the variables s, t, u etc. range over

terms while f, g, ... range over new function symbols or the corresponding elements

of the model.

1.4. Conditions

The notion of condition is formulated more simply than in [19] but the reader of

[19] will easily see the equivalence. The main improvement is that new function

symbols a re not required to be inequivalent. A condition is formally a pair

P = (S(P), == ,,) which satisfies the following.

(a) Properties of S(P) l) S(P) is a finite set of terms.

2) 0, IE S(P). 3) If s ~ t and t E S(P) then s E S(P).

(b) Properties of == , 1) == , is an equivalence relation on S(P). 2) == ,-equivalent terms have the same order.

3) Every term of order - I in S(P) bears =, to exactly one of 0 or 1. 4) If t 1== ,tz, /1 ~ t, and t E S(P) then Sub(t~, t2, t)= ,t.

Adding dependent choice 109

Henceforth the variables P, Q, ... range over conditions.

(c) The ordering on conditions Say P ~ Q when S(P) ~ S(Q) and =P agrees with =0 . Say that P and Q are

compatible when they have a common extension.

1.5. Compatibility Lemma. P and Q are compatible exactly when =, and = 0

agree.

Note. This is the unproved Lemma 3.7 of [ 19]. Its simplified form is due to the

improvement in the definition of condition.

Proof. If P and Q are compatible then = , and = 0 certainly agree. The problem is

to assume that =p and = 0 agree and produce a common extension R. The construction proceeds by downward induction on the finitely many orders, a, such

that S(P) U S(Q) contains terms of order a. The inductive hypothesis is that S(P) and S(Q) contain exactly the same terms of order > a. The inductive conclusion is that there are P *;;;: P and Q *;;;: Q such that:

(a) S(P*) U S(Q*) has terms of exactly the same orders as does S(P) U S(Q). (b) S(P*) and S(Q*) have exactly the same terms of order ;;;: a.

(c) =~,. and =0 • agree. At the last step, a= -I, S(P*) = S(Q*) and = 1,. agrees with = 0 •. Thus

p* = Q * is a common extension to P and Q.

To define P * and Q * let Sa (P) and S" ( Q) respectively denote the set of those

terms of S(P) and S(Q) with order a. On S,, = S" (P) U S, (Q) let = denote

TC(=p f Sa(P), = 0 f Sa(O)). From l.l(b) = is an equivalence relation which agrees

with both =, f Sa(P) and =of S, (Q). p* and Q *are defined symmetrically so only the definition of P *is given. Define

Extend = to a relation ""' on A= A (P) U A (0) by saying t ""' t' if and only if for

some s = s', t =Sub (s', s, t'). ""' is an equivalence relation. The crucial point for transivitivity is that a term has at most one segment of order a. ""' also agrees with

= on S". Notice that ""' fA n S(P) C =I' fA n S(P). If t, t' E S(P) and t ""' t' then the s

and s' in question are both in S(P) by 1.4(a)3). Thus s = ,s' since = , agrees with =. t =i,t' by 1.4(b)4).

Set S(P*) = S(P) U A (P) and let = ,.be TC( = ,, ""' fA (P)) . = ,.agrees with =, by l.l(b). Thus once P* is seen to be a condition it is automatic that P ~ P*.

Consider the definition of condition as applied to P*. 1.4(a)l) and 2) are

immediate as is 3) in the cases~ t E S(P). If s ~ tEA (P) then either sEA (P) or

s < u for some u E Sa. In the latter case s E S(P) by the induction hypothesis.

1.4(b)l) and 2) are also immediate. 0 I= I" I since =~,. agrees with =,.The rest of

110 D. Pincus

1.4(b)3) fo llows since eve ry t EA (P) bears = , he nce = , ., to somes E S(P) US,

and (in case a= - 1) ""' ex tends =o on S".

It re main s to ch eck 1.4(b)4) fo r P*. = ,. is TC( = .:, ""' I A (P)) so, using tran sitivity of = ,. , it suffices to establish Sub(t1 , 12, t) = ,. t under th e special assumptions

11= , !2 and t1 ""' !2. If t1 ""' l2 th en !2 = Sub (s1 , S2, l1) fo r some s1 = s2 in S, . It follows

tha t

If t1= , 12 and IE S(P) 1.4(b)4) can be applied in P. If l1 = ,t2 and 1 E A(P)- S(P) then some s .:; 1 is in S, and I = Sub (s ', s, t') for so me 1' E S(P) . If s E S(P) then

t E S(P ) by 1.4(b)4) appli ed in P. Thus it may be assumed th at s E S(Q)- S(P). t 1 a nd 5 a re bo th .:; I so !1 <S s o r s <S t1 . s <S t1 is imposs ible since 5E S(P) so 11.:; s and 11 has o rd e r > a. l1 and l2 have th e sa me orde r > a so th e induction hypoth esis gives 11, 12E S(P)nS(Q) and t1 = o l2. 1.4(b)4) appli es 111 Q to give Sub(l 1, 12, s) = 0 s so Sub(l1 , !2, s ) ""' s. Th e refo re

Sub(t 1, l2, 1) = Sub(5, Sub(11 , l2, s ), 1) ""' I.

T he proof th a t P*, a nd simila rl y 0 *, a re conditions is now comple te. The indu cti ve conclu sio ns (a), (b ) and (c) mu st still be ve rifi ed . (a) and (b) are immedi ate

since th e o nly new orde r of te rms in P* and 0 * is a . T erms of orde r a only co me int o P* and Q * wh e n S, (P ) U Sa(O) ~ 0.

In pre pa ra tio n fo r th e ve rifi catio n o f (c) obse rve th at if 5* E S(P *)n S(Q *) has

o rde r < a th e re is an s E S(P) n S(Q) such that 5* = s o r 5* ""' s. It may as we ll be

assumed th a t s* E S(P) n S(Q) he nce th at s* EA. Thus the re is an s 1.:; s* fro m

S,. = , is TC( = , , ""' fA (P)) so e ith e r s* E S(P) o r 5* ""' u fo r u E S(P). In the

la tt e r case fo r so mes.,""' s1 Sub (s 1, 52, s* ) = u E S(P). In e ith er case, a llowing fo r

s1 = s2, there is 52 with S ub( s~, s2, s *) E S(P) and s1 = Sz. Similarly the re is s1 = s3

with Sub (s 1, 5_, , 5*) E S(Q ). 5z E S(P) and s-' E S(Q) so by 1.1(b) th ere is an

s~ E S .. (P) n S, (Q) with Sz= l's~ = o s 2 . Set s = Sub(s ~, s4, s* ) ""' s *. Applica tions o f I .4(b)4) in P and Q give , as d es ired:

s = Sub(s2, s4, Sub(s 1, s2, s*)) E S(P)

= Sub(s,, s4, Sub(s1, s_,, s *)) E S(Q).

A t las t indu cti ve conclu sion (c) can be ve rifi ed . By symm etry of hypoth eses it

suffi ces to assume s="· I fo r s, I E S(P * ) n S(Q *) and prove s =o· 1. This is th e ind ucti ve hypo thesis if s, I E S(P) n S(Q). It is a lso clear if o rde r s= o rde r 1 ;;;,: a . O th e rwise suppose by sy mmetry o f s, I th at s E S(P) n S (O) and conside r first th e case 1 E S(P) n S(Q). Fro m above th e re is an s' ""' s, s' E S(P) n S(Q). s'= ,l so the indu ctio n hypo thesis gives s'=o t. Also s ""' s ' and bo th are in S ( Q *) so s =0. s'. Therefo re s =o · t. T o handle the case t~ S(P) n S(Q) introdu ce a 1' ""' 1, t ' E S(P) n S(Q). Th e above a rguments g ive s= o· s'=0 1'=0.t so s =0.t.

Adding dependent choice Ill

1.6. Countable antichain condition

Any set of pairwise incompatible conditions is countable.

Proof. This fact is not needed in the proof of Theorem 1.1. It is of interest since it implies that every real of U(c:9) lies in .N where c:9 is a generic set of conditions. It is also needed to prove preservation of cardinals where the iteration is carried past w 1•

It follows from the general considerations of [23]. Here is an elementary proof. Let 1[1, be the map S(P) x S(P)__, 2 given by o/p (s, t) = 1 ~ s ='=pt. Since the set

of terms has cardinal w, , o/p can be thought of as a Cohen condition for adding a subset of w~, [1]. 1.5. implies that P and Q are compatible if and only if.o/p and 1[10

are compatible Cohen conditions. Cohen conditions satisfy the countable antichain condition.

1.7. Forcing

Forcing for the model is determined once it is decided which constants are strongly forced to be members of new constants. Strong forcing is denoted by II-* while weak forcing is denoted by II-.

(a) Pll-* "cEf"~(3f,g,f3)[PII-* "c=({3,g)"11[{3==,g] (b) P II-* "c E I" ~ (3 f) [ P II-* "c = f"].

1.8. Restriction lemma

(a) Forcing automorphisms Let 7T be a permutation of new function symbols satisfying order 7Tf =order f.

The action of 7T is extended to terms via 7T (ff3, · · · {3 .. ) = ( 7Tf)f3, · · · {3., and to constants by letting 1r(/) =I and otherwise following the recipe of [IJ. 7T acts on

conditions through 'its action on terms. It is standard that if C/J(x, · · · x.,) is parameter free then

p II- C/J ( c, · · · c,.) ~ 1rP II- (p ( 7TC, . . . , 7TC,. ) .

Such a 7T is called a forcing automorphism.

(b) Restrictions of conditions Let F be a finite set of new function symbols. P f F is defined as follows.

S(P f F)= 2 U {t E S(P): (3[ E F)(f ~ t)}

==I'IF = ='=p f S(P f F).

P f F is a condition, P f F ~ P, and P ~ Q ____, P f F ~ 0 f F.

(c) Homogeneity lemma Let P f F and Q f F be compatible. There is a 7T which acts as the identity on F such

that P and 1rO are compatible.

112 D. Pincus

Proof. Notice that this is the unproved Lemma 3.10 of (19] . The strengthened form

of the compatibility lemma re nde rs th e proof much easier than the sketch of (19].

Let 7T fix F and map each f E S(O)- F outside of S(P). S(P) n S( 1rO) c S(P r F) U S(O r F) so th e compatibility le mm a from subsec tion 1.5 proves that p

and 1rO are compatible .

(d) Restriction lem ma Let C/> have parameters in U U {I} U F. P II- C/> ~ Pr F II- C/J.

Proof (1] . If some Q ~ P r F sat isfi es 0 11- -C/> the n so does 1rO for every 1r fixing

F. If 1T is as in th e homogene ity le mma th e n a common ex te nsion to 1rO and p forces a contradiction.

1.9. The model

The a bove se tup leads in (1] to a model}{, U( CS ) ::J }{ ::J U where C§ is aU-generic

se t of conditions. Cofinalities, hence cardinals, of U are preserved in U(CS) and }{

by 1.6. j{ == U[I], the smallest transitive submodel of U( CS ) containing U and the

tran sitive closure of I . (L : a < w 1) is defined in }{ from I by setting Io = In zw, I,+ I = I n !';:, and

I .. == In (U [J <a i13 )" for limit a. Standard arguments using the compatibility lemma

show that /.., = {f : / has order a} , each f E /,.+ I maps w onto /"'each f E 1", a limit , is a choice function from (113 : f3 < a). Furthermore Pll- ff3 1 · · · {3. =

gy 1 · · · y1 ~ ff3~ · · · {3. = /'gy1 · · · y, a nd P II- ff3~ · · · {3. 1- gy1 · · · y, ~ f/3 1 · · · {3. Fl' gy 1 · ·. y1 while both te rms are in S(P). A similar re mark holds for equations

between ff3~ · · · {3k and 0 or I.

1.10. Internal perception of forcing

The remarks connecting truth and forcing in 1.9. were made from the external

viewpoint of U( CS ). It is also important to refilize what can be said about truth and

forcing from the standpoint of }{. The constants, conditions, and forcing relations

are defined in U, hence in .}{, The relationship be twee n these constants and the

elements of }{ must still be torma!ized .

(a) Satisfaction (in .}{) Let p be a condition involving the new function sy mbols /~, ... , / •. (Recall the

conventions of 1.2. ). Let f1 , . . . , f. E I. f~, ... , f. satisfy P (for/~, ... , /.)just in case:

(1) f; E I" ~ f; has order a, each i = 1, . .. , n.

(2) For eac h }J3, · · · /3 k, hY1 · · · y, E S(P),

[;{3 1 ' ' ' Pk = /;)'I ' ' ' '}'t ~ /.{3 I ' ' ' f3k =: P h'Y I ' ' ' )'t

f,f31 ... {3. = 0 ~ f,f31 . . . {3. =I' 0

[;{3 I ••• P• = 1 ~ J,f3 I ... {3. = p 1.

Adding depende11t choice 113

(b) Satisfaction Lemma (in X ). Let P involve J,, ... ,/"g' · · · g'". Let f, · · · f" satisfy P.

Proof. The proof is carried out in U and uses the connection between truth and forcing in U(C§). Everything which appears is either a constant or a constant for a constant. Instead of using double dots simply assume that each symbol has one less dot than it "deserves".

If the lemma is false it must be forced false for p by some Q :;;;. p r {f, .. . fn} such

that Q = Q f {f, · · · f"} . P and Q are compatible by the compatibility lemma. If R is a common extension R forces f, · · · f", g, · · · g'" satisfy R hence P. This contradicts

the choice of Q. Let cfJ(x, · · · x") have parameters in U U {I}. Say that P forces cfJ (relative to X)

if cfJ(/1 • • • fn) holds for every f, · · · f" which satisfy P.

(c) Truth Lemma (in X). Let cP(x, · · · x.,) have parameters in U U {I}. Assume that cfJ (f, · · · fn) holds for f, · · · f., E I. There is a P involving exactly f, · · · f, such that f, · · · f" satisfy P and P forces cP (in X ).

Proof. Clear from [1].

(d) Local intervals If a is a successor or 0 a local interval in Ia is a set of the form {f E Ia :

f(n1) = g,11 · · · 11 f(nk) = gd for some n, · · · nk E w, g, · · · gk E Ia-I · For limit a a local interval in Ia is a set of the form {/ E Ia : f(f3,) = g, 11 · · • 11 f({3k) = gk} for

{3 1 · · • {3k E a, g, E 113,.

Each local interval is nonempty by the satisfaction lemma. Each is actually infinite since it necessarily includes 2 disjoint nonempty local intervals (hence

4 etc.) .

(e) Continuity Lemma (in X ). Let cP(x, · · · x.,) have parameters in U U {I} U

U l3 < a I13 • Assume that cfJ(f, · · · f .. ) holds for distinct f, · · · f., E I a . There are disjoint local intervals J, · · · 1 .. C I , such that [. U J, i = I , ... , n, and cP(f; , .. . , f;,) holds whenever f: E J,, i = 1, .. . , n.

Proof. Assume a is 0 or successor. A similar argument works for a limit. By the truth lemma there is a P which forces, in X, that cfJ holds and is sa tisfied by the [t and parameters from U l3 < a 113• There are n, · · · nk E w such that some term of the

form [. (n;) E S(P). Let J, be the interval {f' E J, : f'(n,) = f,(n,) 11 • · · 11 f'(nk) = {; (nk}}. Contract the J, further to make them disjoint and still contain{; (the{; are

distinct). If f: E J,, i = 1, ... , n then the f: satisfy P.

1.11. The support structure of X

(a) Definition and basic properties A support is a finite G C I. G supports x if x is definable in X from parameters in

114 u. Pincus

U U {!} U G. 'VG is the class of x supported by G . .N = U(I] so every x E .N is in some 'VG.

Each L. has a canonical linear ordering obtained inductively by considering I"' as

a subset of/';:_, or (Uil <a iil)"' and using the least difference ordering. Therefore each finite G c I has a canonical well ordering. This induces a well ordering on

definitions from parameters in U U {/} U G (recall that U satisfies E). And hence on the elements of 'VG, T is defined from the parameters U and I.

(b) Level supports G C I is level if G C I"' for some a. Every x is in \1 G for some level G. In fact if

G ;i 0 and x E \1 G then x E \1 G n I"' for the largest a satisfying G n I"' ;i 0. To see

this notice that iff E I"' and g E Ill for {3 <a then g = fy, · · · /'k for some sequence

of y's.

(c) Intersection lemma (in .}{) If G,, G2 C I,. and x E 'VG, n G2 then either: (1) G,nG2;i0andxEV'G,nG2 (2) G, n G2 = 0 and either x E \10 or for some G C Ill , {3 <a, x E \1 G.

Proof (Essentially (19] Lemma 3.16). Suppose G, n G2 = {/1 • • • fd . x = T(g, ···g . ., f, · · · f~, y) = T(f, · · · fk, h, ···h .. ., 8). Let I,, ... ,/'" be local intervals in­

side which T(g;, ... ,g~,f,···fk,y)=T(f,···fk,h,···hm,8) holds. {x}= {T(g ;, ... , g ~,f, · · · fk , y): g; E I,} and x = U{x }. This definition of x has parameters

from G, n G2 and the local intervals I~, ... , I.. If G, n G2 ;i 0 any f, in the

intersection defines I~, ... , I •. Otherwise the I~, ... , I" are the only essential parame-ters. These are defined in terms of U or G C Ill, {3 < a.

(d) Canonical supports (in .N) The intersection lemma shows that for every x there is a canonically determined

(from I and U) G. with x E G •. G. is found by letting a. be the least a with

x E G C I"' and letting G. be the G CIa. of minimal size with x E \1 G.

Proof of Theorem 1. (a) Proof of - C~' in .N.

(I,. :a < w1) is an w,-sequence of countable sets. Assume T( G, y) is a choice

function for the I"' where G C Ill. Consider the formula C!J(x) ~ "x = T( G, )', Ill+ 1)".

The continuity lemma implies that if $(/) holds then $(!') holds for every f' in some interval containing f. (b) Proof of 0 in .}{.

The canonical linear ordering of I induces one on U a<w, <!P <w (I"' ) where <!P < K (x) denotes the subsets of x with cardinal < K. Let {3. be that {3 with x = T( G., {3.) (see l.ll(d)). The map x ~ (G., {3. ) embeds the universe into (actually onto) the ordered class (Ua <w,<!P<w(I"'))xOn.

This argument actually proves KW (see Glossary) since the elements of Uo <w,<!P<w(I,.) can be coded in the reals.

Adding dependent choice liS

(c) Proof of DC in K. Let R be a relation satisfying the hypotheses of DC. Let x E Domain R. A

designated successor of x is a y = T(g, y) where XRy, g E JIJ for {3 least possible, andy is least possible for the given {3. Note that T({g }, y) is simplified to T(g, y ). It is always possible for g to be a singleton by passing to a higher la.

x has at most countably many designated successors. To prove DC for R define

sets X" and ordinals a" so that Xo = {xo} for some Xo E Domain R, Xn +t = {y : (3x E X" )[y is a designated successor of x ]}, and a" is the least a such that

('V x E X" )(3 G C IIJ )[ x E V' G]. Let a = Sup a" and f E Ia. f defines a well ordering of u(J <ag><w (I(J)) X On, hence of UnewXn. The sequence through R lets Xo be as

given and Xn+t be the least designated successor of x" in the well ordering of X". This completes the proof of Theorem 1. Before concluding this section I will

discuss the problem of proving PI in the model, and thereby adding DC to the independence of AC from PI.

My plan of attack is essentially that of [8]. This argument gives rise to a combinatorial conjecture which is stated below in 1.15. The conjecture is actually a sequence (Ca : a < w.) of conjectures of increasing complexity. My claim of [18] was based on an incorrect proof of the conjecture. This proof is correct for C., hence for Co. The combinatorial theorem of Halpern and Lauchli [7] , arising from

the argument of [8] in the case of U[Io], is intermediate in strength between Co

and c .. The proof of c. will appear separately along with an interesting reformulation of

the theorem of [7]. This reformulation is independently due to R. Laver.

1.12. New notation on conditions

Let F(P) denote the set of new function symbols of S (P). If CT is a 1:1 map F ~ F' where F C F(P) and F' n F(P) = cp let Sub (CF, P) be the result of replacing f by CTf throughout P. When CT is understood Sub (CF, P) will be written Sub (F, F', P). Sub (F, F', P) is evidently a condition as is Sub (F, F' , P) f F' . This last condition will be written Sub IF, F', P 1.

P is a normal condition when no two function symbols are ?-equivalent and any two ?-inequivalent terms of order ~ 0 have witnesses (i.e.

[ t 1, t2 E S (P) td.Ji Pt2] ~ (3{3 )[ t.{3, t2{3 E S (P)" t .{31 t2{3 ]). It is straightforward using the compatibility lemma that if P satisfies the first part of the definition of

normal then some Q ~ P is normal. (Add new t{3's from the bottom level up).

1.13. Normal sequences and matrices

The sequence (P; : i E w) of conditions is normal if:

(a) Each P; is a normal condition.

(b) There is a single a such that each F(P;) contains only function symbols of

order a.

116 D. Pincus

(c) JF(Po)J=1 and i;ij-'?F(P,)nF(PJ=0.

Iff E F(P,) and g E F(Pi) fori~ j say f ~ g if P, f f ~Sub Jg,f, Pi J. (Note the dropping of curly brackets in the above notation.) ~ is clearly a partial ordering . u is understood to take g to f.

(d) (Vg E F(P< +I))(3fE F(P,))[f~ g]. (e) (Vf E F(P,))(3g E P, ~ ~)[f ~ g ]. Clauses (c), (d) and (e) say that U, F(P,) forms a perfect w tree under ~ and

F(P,) is the i th level of the tree. Let r denote the set of successors off in the tree. M c F(P,) is a matrix if there are j ~ i, f E F(Pi ), and a l :1 onto u: M---'? r such

that Pj+l rr ~ subJM,r , P, J .

1.14. Th e conjecture

Let (P, : i E w) be a normal sequence of conditions. There is a j such that whenever F(Pi) is partitioned into two, one part includes a matrix .

C. is the conjecture for sequences with function symbols of order a. (See 1.13(b)). Co and C1 are known. C2 is open .

1.15. The conjecture implies PI in .N'

The argument follows [8]. Also see [21]. If PI is false in .N' it may be assumed [21, Postscript 2] that the failure occurs for a @J = (B: v, 11, --,, 0, 1) E V'0. Let .1' be an ideal for @J which is maximal among those in V'0. Since .1' is not prime there is an

f, 'Y such that { T(f, 'Y ), --, T(f, 'Y )} C @J - .1'. Abbreviate T(f, 'Y) by f Let Po= Po/ f lf-{f,--, f} C @J - j.

The algebraic argument of [8], using the maximality of .1', produces a normal sequence (P, : i E w) of conditions such that whenever /1, . . . , [n, satisfy P, and

M C {/~, ... ,/".} is a matrix then both

(a) A/ eMf E .1', (b) 1\/ eM--,JE.f'.

The contradiction 1 E .1' follows from the Boolean equations

1 = A (h V --, t) = V /\ [i II /\ --, t, j - 1 <M - R uS f ;ER f; E S

where i is taken as in the conclusion of the conjecture. The second expression for 1 is in .1' because each term of the disjunction includes a subproduct of type (a) or (b).

I conclude the section with a word on what happens when the iteration is carried out to some regular K > w1. Cardinals of U are preserved by a remark of 1.6. The proofs of - c::,, 0 , and DC all car~} over directly. oc·· is actually tru e . 1.6 guarantees that .N' contains every set of ordinals in U(<§) whose cardinal is < K.

Notice that this method of adding oc<•, unlike that of Section 3, always de­stroys Cw.

Adding dependent choice 117

2. More applications of the basic method

Theorem 2.1. DC can be added to the following ZF independences. (The resulting models satisfy Z(w ).) (A) 0 from C<w· (B) KW from 0 . (C) OE from 0. (D) Cm from Cnl II ... II en. when S(n,, ... , nk, m) fails. (E) C~0" from Cnl II • .. II c.. when M(n~, ... , nk, m) fails.

The ZF independences of (A), ... , (E) appear as follows. (A) is due to Lauchli [10] in a Fraenkei-Mostowski model and is transferred to ZF in [17). (B) is due to Mostowski [16) in a Fraenkel-Mostowski model and is transferred to ZF in [3]. (C) is due to Mathias [13), (D) is due to Gauntt [5), also see [24). (E) is due to Mostowski [15) in a Fraenkei-Mostowski model and is transferred to ZF in [17). In retrospect results (A), ... , (D) can all be looked at as the results of a single process. A Fraenkei-Mostowski model based on a universal homogeneous structure followed by a ZF transfer as in [17). Results (A) and (D) were looked at in this way by [20] and result (C) by [9]. Only result (A) will be considered in detail here. Result (E) is slightly different and an alternate method of adding DC to this result is given in Section 4. There the model doesn't satisfy Z(w ).

The question as to whether Z( w) ~ 0 was posed by Sageev in [22]. Its plausability stems from the following two facts. Z( < w) is quite weak in Fraenkel-Mostowski models but implies AC in ZF. SP, which is equivalent to KW and hence implies 0, nearly follows from Z(w) and C<w· The only size set which doesn't have a canonical proper subset is a set of size w. Result (A) proves the independence of 0 from Z(w) 11 C<w· Result (E) proves the independence of 0 from Z(w) and SP for families of infinite sets.

2.1. Finite choice operators

The reformulation, given in [20] of Lauchli's result [10] is recalled . The theory of finite choice operators is a first order theory whose relations are equality and, for each nEw, an n + 1-ary relation F". The axioms say that F" is a partially defined n-ary function whose domain is the set of n -tuples with distinct terms. Another set of axioms says that F" (x,, .. . , Xn) is some x,. The last set says that the value of Fn (x 1, •• • , x") is invariant under permutations of the X;. The following are noted. (a) There is a model of the theory with 3 elements a, b, c and isomorphisms

7Tt : [a, b] ~ [ b, c], 1T2: [ b, c] ~ [a, c]

such that 7Tt(a) == b, 7T,(b) == c, 1T2(b) == c, 1T2(c) == a. The square brackets denote generated submodel. (b) The models of the theory satisfy the usual conditions (finite amalgamation and

11 8 D. Pincus

direct unions) for the existence of a countable universal homogeneous structure (with respect. to finite substructures). Fix one such structure~= (A;=, F2, F

3, ••• ).

(c) If 1r is an automorphism of 21" which pointwise fixes H1 n H 2 for finite

H~, H 2 C A then 1r factors 1r = 1Tk • • • 1T1 where each 7T; is an automorphism which fixes either H 1 or H2 pointwise.

Lauchli's model is the one based on the ideal of finite sets and the automorphism

group of 21" . (See the introduction to Fraenkel-Mostowski models in Section 4.1.).

The Cohen model should be viewed as U(Io][Ao][~lo] where U[Io) is as in Section 1, Ao is a generic set of disjoint subsets of Io, and 21"o is the generically added structure

of (b) above. Adding DC is done by a "pushing out" process similar to that of

Section 1. Since A is disjoint every f E I,. adds AC to

U(I13 : {3 < a)(A 13 : {3 < a)(9f 13 : {3 < a).

After adding I,. one proceeds to add A a and ~{ ,. so as to recreate the original model.

Parts (B), ... , (E) of Theorem 2.1. are similar. In (B)~{ is just the rationals with

< . In (C)~ is a universal homogeneous model for the theory of a partial order and

a linear order. In (D) W is the relatively complicated structure introduced in [20). In

(E) A is an m -sized set and W is a cycling successor function on A. In (D) one must

extend U" < w, W,. so as to handle "mixed" finite sets over the A,. 's . It is plausible that these methods also apply to the results of [4) and [24).

2.2. Constants and terms

The new constants are those of Section 1, the elements of A x w 1 (the pair (a, a) E A x w 1 will be written a,. ), sll., fi;, 2 ~ i < w, and ff. The constants are

arranged so that new function symbols of order a are potential members of a,., a,. is

a potential member of sll., i + I tuples of aa ' s are potential members of !Y;, and fi; is a potential member of :Y.

Terms have the same meaning as in Section 1.

2.3. Conditions

A condition is a pair P = (P, PP) where P is a condition as in Section 1. Pp has the following properties.

(1) 1/'p maps S(P) into A x w1. (2) 1/'p(t) E A x {Order t}.

(3) tl = i>l2~ 1/'p(tl) = 1/'p(t2)·

P ~ 0 when P ~ 6 and Po extends Pp.

2.4. Compatibility Lemma. P and 0 are compatible exactly when P and 6 are compatible and 1/'p agrees with 11'0 .

Adding dependent choice 11 9

Proof. The proof of 1.5. shows that if P and 6 are compatible they have a common extension R such that every t E S(R) bears '=ii to an element of S(P) U S(6) and if t E SCR) bears '=ii to elements of both S(P) and S(6) then it bears '=ii to an element of S(P) n S(6). All that has to be noticed is that these two properties are preserved from P to P *. This is explicitly mentioned in the course of the proof.

It is now clear that if P and 6 are compatible and lJI"p agrees with lfl"0 then rule 2.3(3) uniquely extends lJI"P and lJI" 0 in a well defined way .

2.5. Countable antichain condition

One can use 2 U A as one used 2 in 1.6.

2.6. Forcing

Membership in new function symbols and I is exactly as in 1.7. These can be thought of as stipulations (a) and (b).

(c) P II-* "c E cg"" when (3f)[P II- * "c = f" 1\ 1f/p(f) = a" ]. (d) Pll- *"c E~" when (3aa)[P II-*"c = aa") (e) P II-* " c E gj" when (3a )(3( a 0 , • •• , a i) E F; )[ P II- * "c = (a ~, . .. , a ~)''].

(f) P II-* "c E g" when (3i)[P II- * "c = g j "] .

2.7. Restriction lemma

(a) Forcing automorphisms Let C(} be the group of automorphisms of 1.8(a). CfJ acts as in 1.8(a) on new

function symbols and is the identity on other new constants. Let 'Jt = .1w• where !J is the group of automorphisms of ~. 'Jt acts on a" through the action of the a th component on a. It acts as the identity on new function symbols and other new

constants. C(} x 'Jt is the group of forcing automorphisms its action is determined by the

above. Un conditions this means 1Jf,.,( 1rt) = 1r(lf!",(t)). As usual

P II- C/J ~ 1rP + 1re/J.

This is because P II-* "c E g j" is preserved under 1T since !J is the automorphism

group of~-

(b) Restrictions P t F = (P t F, lJI"p t S(P t F)) . The properties of 1.8(b) hold .

(c) Homogeneity Lemma. Let P t F and Q t F be compatible. There is a forcing automorphism 1r, the identity on F, such that P and 1rQ are compatible.

Proof. Let the CfJ component of 1T be as in 1.8(c). Do a similar thing for the 'Jt component using the universality and homogeneity of 2(.

120 D. Pincus

(d) Restriction Lemma (See 1.8(d)). Let cP have parameters m F U (A x w,) u {I, .:1}. If P II- cP then P f F IHP,.

2.8. The model

A cofinality preserving model JV is obtained. The sequence (/, : a < w 1) is defined as in 1.9 . .rA. is a partitioning of I and fJ; is an i + 1-ary relation on .rA.. The sequence (;!11 : i E w) is definable from g = {fJ; : i E w} . .rA. is also defined from fY,

say as the domain of fYz, so every set in JV is definable from parameters in U U {;!!} U G U H where G C I is finite and 'Je C .rA. is finite.

A . = .rA. n ~ (/, ).~I " = (A, : fYz fA,, .o/"3 fA, · · ·)is defined in .!V and is, relative to JV, a countable universa l homogeneous model of the theory of finite choice operators. An enumeration of A. can be defined from any f E /,+ z·

2.9. Internal perception of forcing

(a) Satisfaction fz, ... , fn E I and az, ... , am E A satisfy P if

1) fz, .. . , fn satisfy P. 2) For each a the map a,~ a, is an isomorphism of the structures [A, n {az, . . . , am}] and [A X {a} n {ciz, ... , cim}]. A X {a} inherits a structure via the map

(b) Satisfaction Lemma (See l.lO(b )). Let P involve j~, .. . , jn, gz, ... , gk, a~, ... , am, b~, . . . , 6,. Iff~, . .. ,f,., az , ... , a, satisfy P there are gz, ... , gk, bz, .. . , b1 which satisf"y P

together with f~, ... , fn , a z, • .• , a,.

(c) Truth Lemma (See I.IO(c)). Let cP(xz, .. . , x,., y~, . .. , Yn) have parameters in U U { .o/" }. If cP(f~, .. . , f.., a,, .. . , a,) holds there is a P satisfied by fz, ... , f.., a 1, •• • , a'" such that cP(f; , .. . , f;,, a;, ... , a,;,) holds whenever f; , .. . , [:., a;, ... , a:.. satisfy P. (P

forces (p in JV. )

(d) Local intervals (Exactly as in l.lO(d))

(e) Continuity (See 1.10(e)).

Let W(x 1, •• • , Xn, Yz. ... , Ym) have parameters in U U {.o/"} U Ull<aiil U Ut~ < a Ail. Assume cP(fz, ... ,fn, a~, ... , am) holds for distinct f. E I"', ai E U "'"'llAil. There are disjoint local intervals Jz, .. . , In C I, containing f~, ... , [n such that cP(f;, . .. ,f~, a;, ... , a:,) holds whenever:

(1) [: E I; i = l, ... , n (2) a;~ ai is an isomorphism of [All n {a;, ... , a,;,}] and [All n {a~, .. . , am}] for all {3 < Wz.

(3) J: E aj~ f; E ai.

Adding dependent choice 121

2.10. Support structure

(a) Definition A support is a pair G, H where G C I is finite and H C .s4 is finite. G, H supports

x if xis definable in}( from parameters in U U {.o/"} U G U H. VG, His the class of

x supported by G, H. Every x lies in some 'il G, H. Each G has a canonical well ordering as in 1.11 (a). H also has a canonical well ordering . First order according to

the a for elements in different H n A .. In a fixed H. = H n A. let the first element

be F /l·l., l (H.), the second be F 1H. I- 1 (H. - {FrH .. I (Ha )}), etc. In terms of this well ordering let T(G, H, {3) be the {3th element of VG, H.

(b) Normal supports G, H is called normal if for some a, G C I,., H C U a .. 13 A 13 , and

(VfE G)(3a E H)[fE a]. For every G, H there is a normal G*, H* such that

VG, H = VG*, H*.

Find G* as in 1.11(b) and let H* = (H n Ua., 13 A13 ) U {a E Ia: (3/ E G*)[/ E a]}.

(c) Intersection Lemma. Let G~, H 1 and G 2, H 2 be normal supports with G~, G 2 C

Ia. Let

x E VG~, H, n VG2, H 2.

1) If G 1 = G 2= G then x E VG, H, n H 2.

2) If H, =Hz= H then either: a) G,nGz-10andxEVG,nGz,H b) G 1 n G2 = 0 and there are normal G', H', and such that G' = 0 or G' C !13 , {3 < a

X E'ilG',H'.

Proof. Case 2) follows as in l.ll(c). Case 1) will look familiar to readers of [9] (see

the "support theorem"). G, H, and G, H2 are normal so

{a: (3f E G)[f E a]} C H, n H2. x = T(G, H~, 1) = T(G, H2, S)

for some ordinals y and S. As in 1.11 (c) it turns out that one can define x = U{x}

and

{x} = {T(G, H' , y): [H'] ~ H, nH, [H,]}.

[H'] ~ H, nH, [Ht] means that there is a map a-: H' ~ H, such that a- is the identity on H 1 n H 2, a E A 13 ~era E A 13 , and for each {3, a- f H' n A 13 is an isomorphism of [H' n A 13 ] onto [H, n A 13 ]. The only essential parameters of the definition are those of G and H, n Hz since the isomorphism type of [H, n A 13 ] over H, n H 2 can be described from parameters in u. Therefore X E 'il G, H, n H 2.

It remains to show that the above expression really defines {x}, i.e. that x = T(G,H' , y) whenever [H'] ~ H, n H , [H,]. Accordingly let H' and a- be given.

Let cp embed B = H' U H, U H 2 into A x w, in such a way that cp is an isomor-

122 D. Pincus

phism from each [B n A.] into 91 x {a}. Lett/', H,, H z, etc. denote the respective

images of H', H~, Hz, etc . under cp. Using the homogeneity of 91 there is an extension 7T of cpacp _, to an automorphism of 9! x w,. 7T is the identity on H, n Hz

so by 2 .1 (c) 7T factors 7T = 7T" · · · 7T, where each 7T; is the identity on H, or f/2

•

(Note: This property of 91 easily extends to a finite power of 2L) Let Hi or H~ 0 0 0 0 0 0 0

denote 7Ti · • · 7T,[H,] (or H z). Evidently H, = H; and H';' = H'. Let fi = u;_o H f u H ~ . The satisfaction lemma (2 .9(b )) gives an H :J H, u H z and an

extension of cp, also called cp, which maps [H] isomorphically onto [H]. Let H{ (or

HO denote cp - '(H:) (or H 0. It is claimed by induction on j that x = T(G, H{, -y) =

T( G, H ~' o ). Setting j = n gives the desired conclusion x = T( G, H', 'Y ).

The induction is carried out as follows . Assume, for the sake of definiteness, that

7Tj +l fixes H z pointwise. By the inductive assumption T(G, Hi, -y) = T(G, H2

, o). cp - 17Ti+I'P applied to Hf U H z fixes H z, hence H, n H z, pointwise and is otherwise an isomorphism. By continuity (2.9(e))

T(G, HI +' , -y) = T(G, 'P - 17Tj+I'P[H{], -y) = T(G, cp - 17Ti+I'P[Hz], o)

= T(G, Hz, o) = X.

A similar argument , applying 'P - 17Ti +' · · · 7T,cp to H, U Hz, gives the conclusion

T(G, H~· ' , o) = T(G, HI+', -y) = x.

This completes the induction and the proof.

(d) Canonical supports (in JV ) There is a definable (from U , :Y ) function which takes x to a normal support

G., Hx such that x E VG., Hx.

Proof. Let a x be the least a such that for some normal G C I" and H , x E VG, H. Let Gx C !,_,, be such a G of minimal size. Gx is unique by case 2) of (c) above. If H works for G, and G ' , H' are given with x E VG', H' and G C la, then x E VG' n G., H' U H.

Now fix Gx and apply case I of (c) to obtain a minimal H, such that G., H. is

normal and x E V G., Hx.

2.11. Proof of Theorem 2

(a) Proof of DC in JV (see 1.12( c)) .

(b) Proof of C<w in JV. Let X be a given finite set. Set G = u xEX G., H = u xEX Hx (See 2.IO(d)).

Choose from X that element which is T( G, H, a) for the least a.

(c) Proof of Z(w) in JV. Let X be a given infinite set. Since w, is regular there is a least a < w, such that

the intersection of X with the union of the VG, H, G c U 13 < " !13 , H C U 13 < " A13

is

A.dding dependent choice 123

also infinite. This intersection is well orderable and is the desired countable subset

of X if it is countable.

If the intersection is uncountable there is, by DC, at least one G C U i3 <a ! 13,

H C U 13 <., A 13 such that 'VG, H is uncountable. Let X c.H be the first w many

members of X n 'il G, H under the ordering induced by T. The desired set is now

taken as the union of the X c. H over those G C U i3 <a ! 13 , H C U i3 <a A 13 such that X n 'VG, H is uncountable.

(d) Proof of - 0 in }{ There is no ordering of s4. in K. Suppose < = T( G, H, y) is such an ordering

where G, H is normal and H C U i3 <a !13• It suffices to show that < does not order

A.,. The universality of ~1 implies that the structure [a, b, c] described in 2.l(a) can

be thought of as a substructure of A x {a}. The satisfaction lemma (2.9(b)) provides a substructure [a, b, c] of 2( a and an isomorph ism if' taking a to a (resp . b, c).

Assume a < b i.e. (a, b) E T(G, H, y). Continuity using a map which is the

identity on Hand is cp - 11T1if' on {a, b} (see 2.1(a)) gives the conclusion b < c. A

similar argument using b, c, and 1T2 gives c < a. This is a contradiction. The same

contradiction follows from the assumption b < a. Therefore < is not a linear

ordering of A .,.

3. Adding nc<•

Let K = K" be a fixed regular cardinal in the ground model U of ZFE. (K <• is the

set of < K -sized subsets of K.)

Theorem 3.1. oc<• can be added to the independence of AC from 0 and c < On·

Theorem 3.2. C. is independent of C<. in ZF.

These results are subject to the usual problem of the meaning of a statement

involving the set theoretical parameter K. Levy [12] discusses this question. The models of Theorems 3.1 and 3.2 preserve the meaning of K with regard to

statements absolute in the cofinality function . In the semantic sense of [12] it should

be stated that both }{'s preserve the cofinalities of U and power sets of U-sets with

cardinal < K.

Theorem 3.2 has a history. Levy [ 11] originally claimed the independence of C.

from c <. for limit K using a Fraenkei-Mostowski model. His proof was incorrect

and Howard [6] subsequently proved that no Fraenkei-Mostowski independence

proof of C<o .. from C <w is possible. Sageev [22] and I [19] gave the first correct

independence proofs of C w from C <w· Theorem 3.2 generalizes this to arbitrary

124 D. Pincus

regular cardinals. This is more than Levy claimed in [11] since his methods left the successor case open. It is also Jess than he claimed with regard to singular cardinals.

The independence of C. from C <. for singular cardinals remains open.

As was implied in the introduction both parts of Theorem 3 differ rather radically in proof from Theorem 1.1. To see why this is necessary consider what a straightforward generalization of the proof of Theorem 1.1 . wou.Jd be.

Cohen's original model U(I .. ] has two natural generalizations to K. In both cases

]0

becomes an independent set of subsets of K introduced by < K closed conditions.

In the first case only finite sized subsets of lo are kept while in the second case < K

sized subsets of Io are kept. At first sight the second model might seem the natural

one to use as DC<" is easily seen to hold there. However the proofs of 0 and C <o.,

both of which hold in Cohen's U[Io] break down when K > w. Set two enumerations

of subsets of [ 0 equivalent when they are eventually equal. The equivalence class of

any 1- 1 enumeration has no minimal support. At first sight the above observations don't seem to present an obstacle to carrying

out th e plan of Section I. Define U[Io] so that only finite sets are kept. 0 and C <o .. hold there . Then let I, be an independent set of maps from K to Io etc. But how,

exactly, is I, to be added? lo is Dedikind finite in U[Io] but it also maps onto K .

(There is a K -sized collection of disjoint intervals.) Thus if conditions talked about

only finitely many members of l o they would collapse K tow. On the other hand, if

they talked about < K-sized subsets of Io they would be talking about sets which

don't exist in U lo. It would quickly follow as in the second generalizataion of U[Io] ,

that canonical supports do not exist. In fact the support intersection lemma fails in

U[lo, I,] because any countable subset of Io will be supported by {f} for any f E J, but will not be supported by a finite subset of I 0 •

The problem is solved as follows. Before adding J, one must add the < K sized

subsets of [0 back in. Furthermore one must do this in such a way as not to lose 0

and C<o .. in the resulting model. Thus one obtains not the second natural

generalization of Cohen's U[ lo] but a larger model where canonical supports and

canonical well orderings exist. Consider, then , the idea of adding in first thew-sized sets, then the w,-sized sets,

etc. until all the < K-sized sets have been added in. Using again the idea of Section

1 think of [0 = Io.o and let lo.1 be a set of maps from w into Io.o. lo.1 will of course be

Dedekind finite in U[ Io][Iru] but this will be removed when Io.2 is added etc. Finally

when UUo.a : a < K) has been formed one will have added back all < K subsets of

10 and be ready to add I,. Still a problem remains with the formation of UUo.a :a < K ). Even the condi­

tions which add elements of lo.1 to U[Io] must talk about infinitely many elements of

fo. Otherwise what is to prevent w from being mapped onto lo? But then why won ' t countable subsets of lobe introduced which don't have a canonical support? This will not be a problem if one stipulates that the sets enumerated by elements of ] 0 .1 are almost disjoint. Similarly for higher Io .a 's a notion of hereditary almost

disjointness will be formulated so as to prevent sets from coming in without

Adding dependent choice 125

canonical supports. It should be remarked that complete disjointness won't work

because in that case UUo.a : a < K) would satisfy A C. All of the above work went to permit the introduction of It. Similarly I .. can be

introduced for n < w. However this process won't permit the introduction of Iw because any element in the appropriate J1 will already well order UU ... a : nEw,

o: E K). w seems to be as far as the process goes. The resulting model satisfies C <. and -C. but it fails DC as (I .. : nEw) has no choice function. Thus Theorem 3.2 is obtained but not Theorem 3.1. However backtrack to the model UUo.a : a < K ). This model does satisfy 0, C<o .. , and DC<•. A model for DC<", C <., and -C. continues to be elusive.

3.3. Hereditarily almost disjoint functions

Let X be a fixed set and a EOn. The following notions are defined by induction

on a: (a) hereditarily almost disjoint (HAD) functions of rank a on X ,

(b) an almost disjoint (AD) pair of HAD functions , (c) the values (Va) of the HAD function a, (d) the maximal common values of the AD pair a, b of HAD functions

(MCVa, b). The definition proceeds as follows.

1) The HAD functions of rank 0 on X are elements of X. 2) The HAD functions of rank a + 1 on X are those 1:1 functions from max (w, I a I) whose range is a pairwise AD set of HAD functions of rank a. 3) The HAD functions of limit rank a on X are those functions on a which take each f3 <a to an HAD function of rank f3 and which have a pairwise AD range. 4) If a is an HAD function on X

Va ={a} U U Vaf3. /3 E Do mai n a

5) The HAD functions a and b on X are AD when either Van Vb = 0 or 3c~o . .. , c., EVa n V b such that

n

v a n V b = U V c;. i = l

6) If a and b are AD HAD functions on X MCV ab is either 0 if V a n V b = 0 or is U .. the minimal set of c~o ... , c .. such that V a n V b = i = 1 V C;. The minimal set is

obtained from a given set of C; by throwing away those c; which are in V ci for some jl i. The uniqueness of MCVab is clear.

3.4. Other notions concerning HAD functions

(a) Say a :s.:: b ~a E V b. :s.:: is a partial well ordering . Note: the variables a, b, . .. henceforth range over HAD functions on X.

126 D. Pincus

(b) If A is a set of HAD functions VA = Ua EA Va. Note: the variables A, B, ...

henceforth range over sets of HAD functions on X (c) A is AD if its members are pairwise AD. A is HAD if VA is AD. These

notions will shortly be proved equivalent. Henceforth the term AD will be reserved for sets and HAD will be reserved for functions. The exception to this is when it is

necessary to remind the reader of the equivalence of the notions of AD and HAD

sets. (d) If A and B are sets of HAD functions MCV AB is the unique minimal finite

set , when it exists , of {ct , ... , c.,} such that VAn VB= U~- ~ Vc;. MCVAB = 0 when VA n VB = 0.

(e) If A and B are sets of HAD functions on X and Y respectively an

embedding of A in B is a I: I function <p: VA~ VB such that <p preserves rank

and for every a EVA, {3 E Domain a, (<pa ){3 = 'P (a{3). An onto embedding is an

isomorphism .

3.5. Three basic observations

(a) Existence of MCVAB If A and B are finite and A U B is AD then MCV AB exists. The proof is by

thinning out U MCVab. a E A f, E IJ

(b) Lemma on XA Let XA =X n VA and X,,= X ,,". If A is finite and AU {a} is an HAD set such

that X, C XA then a E VA.

Proof. Let a' E V a be of least possible rank such that a'~ VA. If rank a' > 0 then

one of the (infinitely ma;1y) a'f3 EVa is distinct from all of the (finitely many)

members of MCV{a '}A and has rank =3: any member of MCV{a'}A. On the other

hand the choice of a ' says a' {3 EVA, a contradiction . This means rank a'= 0 i.e .

a' EX, - XA, al so a contradiction. Therefore Va CVA and a EVA.

(c) Cardinalities I X, I= I V, I= max (w, I a I) where a has rank a. There is some use of AC here but

it will be applied only in the ground model.

3.6. Lemma. An AD set of HAD functions is HAD

Proof. Let A be AD (see 3.4(c) .). It will be shown by induction on Supa EA rank a that VA is AD. Let a, bE VA. For some c, dE A a ::;; c and b ::;; d. Without loss of

generality assume c -1 d. Set Bo = MCV cd, B 1 = MCVaBo. (Again note the conven­tion of dropping { }.) B2 = MCVbBo, and B , = MCV B1B2. The existence of the B;

are justified by the AD property of A, the induction hypothesis, and Lemma 3.5. B 3

is clearly a set of common values to a and b. If e is an arbitrary common value it quickly follows that e E V Bo, e E V Bt , e E V B2, and e E VB, .

Adding dependent choice 127

3.7. Amalgamating finite AD sets

Let A and B be finite AD sets on X and Y respectively. Let CCV A n VB be a

finite AD set on X n Y such that (X n Y)c = X n Y . Then A U B is an AD set on

XUY.

Proof. Suppose a E A, bE B . Set A1 = MCVaC, B1 = MCVbC, and C 1 =

MCVA1B1. C1 works as MCVab. Let c EVan Vb. c must be an HAD function on

both X and Y, hence on X n Y. Lemmas 3.5(b) and 3.6 show that c E VC1.

3.8. Span Lemma

Let A be an AD set of HAD functions. Let Y C VA be such that Y C VB for some finite B C VA. There is a unique smallest finite span of Y (Sp Y) such that y c VSp Y and Sp Y C VB whenever B C VA is finite and Y C VB. Furthermore if A CA~, SpY is the same when computed relative to A1.

(a) Sublemma Let CUD U E C VA be finite and such that every element of C U D has rank less

than any element of E. Let a be the greatest rank of an element of CUD and let C.,, and D"' be the rank a elements of C and D. There is a finite F CA all of whose elements have rank <a and

V(E U C) n V(E U D)CV(E U (Can D "' ) U F).

Proof. Set F = (MCVCD U MCVCD" )- (Can D" ).

(b) Proof of the Span Lemma Let Eo = 0 and inductively assume that E1, .. . , E., have been defined such that for

some finite CCV A 1) YCV(E0 U .. · UE., UC) 2) Every member of C has rank less than that of any member of

E = EoU · · · UE.,. Either Y C VE or there is a least a which is the greatest rank of an element of some

nonempty C satisfying 1) and 2). In the first case E .. +I is undefined. In the second case E., +1 is the smallest set consisting of elements of rank a such that some C*

satisfies 1) and 2) with n replaced by n + 1. The existence of the minimal E., +1 is

justified by the sublemma. The sublemma says that if C.,, and D ... are candidates for

E .. +I then so is C" n 0 ,.. The construction of the E., implies that all elements of E., have a fixed order a.,

and that a., +1 <a.,. It follows that there is a largest m such that E"' is defined. SpY

is set equal to E o U · · · U Em. To show that this definition works assume Y C VB C VA and B is finite .

Assume inductively that Eo U · · · U E,. C VB and a E E .. +I- VB. Set Do=

128 D. Pincus

MCV aB and D = Do U E .. +2 U · · · U E .... Every term of D has rank < a. +, and,

since YCVB,

Y CV(Eo U · · · U E .. U (E .. +,- {a}) U D).

E .. +,- {a} is thus shown to be a candidate forE,. +, smaller than the minimal one, a contradidction. Therefore E,. +, C VB and the induction concludes with SpY c VB.

The uniqueness of Sp Y is clear. Two candidates for Sp Y are included in each other's values and their minimality guarantees equality. It is likewise clear SpY is invariant when computed with respect to A. :::J A. Let Y CVB. n VA •. Y CVB 1 n

VA so SpY CVB, nVA CVB •.

3.9. Embeddings and spans

If 1/f and cp are embeddings of A into B which agree on Y then they agree on

SpY.

Proof. 1/f[Sp Y] is clearly Sp (I/![ Y]) computed relative to 1/f[V A], hence relative to B. Similarly cp [Sp Y) = Sp ( cp [ Y]) = Sp ( 1/1 [ Y]) = 1/1 [Sp Y]. Now assume that a, b E Sp Y and V a n Y = V b n Y. a must be equal to b because otherwise YCV((SpY-{a,b})UMCVab) so the same must hold for SpY. aE V(Sp Y-{a,b}) is impossible by minimality of SpY so a EMCVab i.e. a EVb.

Similarly b E V a and a = b. Let a ESpY be given. Since 1/f[Sp Y) = cp [Sp Y) there must be a bE SpY such

that 1/fa = cpb. 1/f and cp are em beddings so 1/1 [V a] = cp [V b]. Therefore 1/f[Va n Y) = 1/f[Va) n 1/f[Y) = cp(Vb) n cp(Y) = cp[Vb n Y). 1/f and cp agree on Y

so V a n Y = V b n Y, a = b, and 1/fa = cpb = cpa.

3.10. Building HAD functions As in 3.5(c) there is a slight use of AC. All applications of the constructions occur

where there is no problem.

(a) Limits of increasing sequences Let (afl : (3 < a) be weakly increasing ( y:.,;;; (3 ~ ay :.,;;; afl ) . There is an a of rank

Supfl <a rank afl such that each afl :.,;;; a and X a = U l3 <a Xaw Furthermore if Y C X and aa =Sp(YnXau ) (relative to aa) for such a then a =Sp(YnVa).

Proof. There is nothing to prove unless y = Supfl<a rank afl is a limit ordinal. In that case there is at most one a13 of a given rank 5 < y. Define as to be that afl where possible and a value of a bigger afl otherwise . The afJ are clearly an AD set so Lemma 3.6 shows that they are an HAD one. It follows that a is HAD. The

other assertions about a are clear.

(b) Dominating a finite AD set with an HAD function Let A be finite. Let a > SupaeA rank a. Let Y C X- XA have cardinal

Adding dependent choice 129

Max (w, I a I). There is an HAD function b of rank a such that A C V b. Further­more if c E Vb- VA then either c;::: a for some a E A or Xb C Y.

Proof. Proceed by induction on a > 0.

Case 1 a = {3 + 1. Let a, , .. . , a" be the elements of A with rank {3. Let A 1 ==A- V{a 1, ••• , a"} . Divide Y dis jointly into Y.,, n :E; y :E; Max (w, I a I) such that I Y'Y I = Max ( w, I a I). Let bo satisfy the induction hypothesis for A 1

, ¥ 0 , and {3. For y == 1, ... , n let b'Y == a'Y. For y > n let by satisfy the induction hypothesis for 0, Y'Y, and {3. The b . just defined is HAD since the Yy are disjoint. The part of the conclusion dealing with c E Vb- VA is also clear.

Case 2 . a is limit. Let y0 < a be the maximal rank of a member of A. Divide Y

dis jointly into Yy, y <a, y nonlimit where I Yy I== Max (w, I y I). For nonlimit y <a let by satisfy the induction hypothesis for 0, Y.,, and y. At y0 + 1 vary this by letting

b(yo+ 1) satisfy the induction hypothesis for A, Yyo+" and yo+ 1. At limit y let (by )8 == b8. It is easy to see that this definition works.

Work on Theorems 3.1 and 3.2 is now ready to begin. Most will be done on the model for Theorem 3.2 since the model for Theorem 3.1 is a submodel of this .

3.11. Constants

The new constants for Theorem 3.2 are a single constant J and K many new function symbols. These new function symbols are assigned orders ( n, a) E w x K . K

many new function symbols have each order. The new function symbols are arranged in the hierarchy so that the potential members off include all pairs (a, g)

where a < K and g has lower order than f. Every f is a potential member of J. For Theorem 3.1 the new constants are the new function symbols of order (0, a)

and the constant l o. Every function symbol is a potential member of l o and the function symbols are arranged as in the model for Theorem 3.2.

3.12. Terms

It is useful to introduce new terms called placeholders. K many placeholders are assigned to each order (n, 0), nEw. G_ray terms are those which begin with placeholders. Others are black. 0 and 1 are both colors. (a) Terms of length 1 are the new function symbols, placeholders, 0, and 1. They have their assigned orders (0 and 1 have order - 1).

(b) If t has order (n, a + 1) its 1 element extensions are of the form t13, {3 < Max (w, I a I). t13 has order (n, a) . (c) If t has order (n, a), a limit, its 1 element extensions have the form t13, {3 < a. t13

has order (n, {3). (d) If t has order (n + 1, 0) its 1 element extensions have the form t", a < K and

have order (n, 0).

130 D. Pincus

(e) If t has the form (0, 0) its 1 element extensions are as in (d) and have order - 1. (f) If t has order - 1 no extensions of t are terms .

3.13. Conditions

For Theorem 3.2 a condition is a triple P = (S(P), =P, H(P)) with the properties listed below. For Theorem 3.1 a condition is a condition of Theorem 3.2 all of the

terms of whose S (P) have order (0, a) or - 1. The ordering is the restricted

ordering .

(a) Properties of S (P) and notation (1) a) S(P) is a set of terms,

b) F(P) is the set of new function symbols of S(P), c) H(P) is the set of placeholders of S(P),

d) Sn,a(P) (resp . Fn,a(P), Hn,a(P)) is the set of terms of S(P) (resp. F(P), H(P)) with order (n, a). H" (P) = Hn.o(P).

(2) F(P) is finite. / S(P)/ < K . Ua<• Sn.a (P) "10 for finitely many n. (3) 0,1 E S(P). (4) If t E S(P) and s ~ t then s E S(P).

(5) If t E S(P) has order (n, a), a "I 0 and t13 is a term then t13 E S(P). Notice that this introduced no contradiction to (2)[3 .5(c)].

(b) Properties of """P

(1) =P is an equivalence relation on S(P). (2) =p-equivalent terms have the same order.

(3) Every term of order (n, 0) or - 1 is equivalent to a unique term of H(P) U 2. (4) If t1=pt2 and t1 ~ t E S(P) then Sub (t1 , tz, t)=pt.

(c) Properties of A (P) (1) A(P) is a sequence (An(P): nEw).

(2) A. (P) is a finite AD set of HAD functions on H" (P). Notice by 3.5(c) that

rank a < K for a E A,(P). (3) H.,(P) = Hn(P)AnU'>· (4) The following associates each t E S(P) of order (n, a) with an a, E An(P) of

rank a . a) a, is the member of H(P) bearing """P to t if t has order (n, 0). b) a,{3 = a,13 if t has order (n, a) , a > 0.

(5) S = pt ~as = a,.

3.14. The ordering of conditions

P ~ 0 exactly when there is a 1:1 map t/1 : S (P) ~ S ( 0) which satisfies the following. (a) t/J is the identity on black terms.

Adding dependent choice 131

(b) t/1 maps Hn(P) into Hn(O) and each t/1 f H"(P) extends (uniquely by 3.9 and 3.13(c)3) to an embedding, also called ifJ, of A .. (P) in A .. (Q) .

(c) t/l(t{3) = (t/lt){3. (d) S = p f ~ t/Js = Q t/Jt. ~ is not, strictly speaking, a partial ordering in that P ~ Q and Q ~ P are

possible for Pf Q. This occurs exactly when Q is obtained from P by a substitution of placeholders. Formally conditions are equivalence classes under ~ . Informally conditions will still be thought of as defined in 3.13. Conditions will, however, freely

be replaced with equivalent ones.

3.15 . Restriction

The proper notion of the restnctwn of a condition is relatively difficult to formulate and must be stated before the compatibilityb lemma.

(a) Weak restrictions p f F, the weak restriction of P to the set of new function symbols F, is defined

essentially as in Sections 1 and 2. S(P f F) is the gray terms of P together with the black terms beginning with an element of F. =,IF is the restriction of=, to S(P r F) and A (P) is unchanged. P f F obeys the usual rules , see 1.8(b).

(b) Strong restrictions

P f F, the strong restriction of P to F, is a further pruning of P j F so as to involve only what is said about elements of F. The black terms of S (P j F) are those of P f F. The gray terms of S ... o(P j F) are defined by downward induction starting with the highest n such that F(P) contains terms of order n. S ... o(P j F)= 0 for higher n. When this induction is complete =, r F will be the restriction of ="IF to S(P j F).A,(P j F)willbeSpH .. (P j F) . Thiswillbeengineeredsoastosatisfy

13(c)(3). 1) The non placeholder gray terms of S ... o(P j F), n ~ - 1, are those of the form

tf3 E S(P) where t is the gray term of Sn +J.o(P j F). 2) The placeholders of S ... o(P j F) are those of Sp (XU Y) where X is the set of

placeholders equivalent to black terms of S ... o(P j F) and Y is the set of

placeholders equivalent to nonplaceholder gray terms of S ... o(P j F) . Sp is com­puted in A. (P). P j F also satisfies the rules of 1.8(b ).

3.16. Compatibility Lemma

Let F = F(P) n F(Q). P and Q are compatible exactly when P j F and Q j F

are compatible.

Proof. It suffices to prove that P and Q are compatible when P j F and Q j F are. For this, in turn, it suffices to show that if P j F ~ Q then P and Q have a common extension R such that R j F( Q) ~ Q. The general case follows from the

special case in 3 steps.

132 D. Pincus

1) Use compatibility of P i F and 0 i F to get a common extension R such

that F(R) = F. 2) Apply the special case to the pairs P, R and 0, R obtaining P' and 0' such

that P' j F ~ R ~ 0'. 3) Apply the special case to P' and 0' obtaining the desired common extension

to P and 0. In the proof of the special case three points should be made about the proof of

the compatibility lemma in Section 1 (1.5). (a) The finiteness of P and 0 is used in two places. The first is to show the finiteness of the common extension. If one only wanted the extension to have cardinal < Kit would suffice that IS(P)I and IS(O)I are< K. The second use of finiteness is to make downward induction on orders possible. For this it would suffice that S(P) and S(O) contain terms of only finitely many orders.

(b) Every tES(R) bears = u to an sES(P)US(O). (c) If t E S(R) bears =R to u E S(P) and v E S(O) then it bears =R to

wE S(P) n S(O). Points (b) and (c) were already discussed in the proof of 2.4.

These facts will be used to build the "gray structure" of a common extension toP and 0. Choose representatives for P and 0 in such a way that the identity function is the !{1 which demonstrates (3.14) that P j F ~ 0 and H(P) n H(O) "= H(P j F). Consider the "gray parts" of P and 0 i.e . P f 0 and 0 f 0. Forgetting, for a moment , the A (P) and A ( 0) structures use the proof of 1.5, together with

observation (a) above, to put together S(R r 0) and =Rrll which satisfy (b) and (c) above. The choice of representatives and property (c) above shows that every term

bears = Rru to a unique placeholder (or 0 or 1). Since P j F is a condition and p j F ~ O, H.,(P j F) = Sp(H .. (P j F))=Sp(H.,(P)nH .. (O)) computed rela­

tive to either A .. (P) or A. ( 0). Therefore amalgamation of AD sets (3.7) applies

and A.(P)UA .. (O) is a finite AD set on H.,(P)UH.,(O). Let A .. (R)=

A.,(R j 0) = A .. (P) U A .. (O). It remains only to put a "black structure" on R. First assign equivalence classes

to the black terms of S(P)US(O) via s=Rt~a, =a,. Since P j F~O, it is irrelevant fortE S(P) n S(O) whether a, is computed in P or 0. The general term

of s ... o(R) is Sub (s~, s2, t) where t is a gray term of R r 0, s I is the initial placeholder

of t and a,. = s 1• Its equivalence class is that of t.

It is routine to check that R is a condition and P, 0 ~ R (identity map). To see that R j F(O) ~ 0 observe that if t E S(O) and t13 E S(R) then tfl E S(O). This is

clear by 3.13(a)(5) unless t E S ... o(R) for some n. Such t bear =R to a gray t' hence one can assume that tis gray. tfl bears =R to a term of S(Pf0)US(Of0). t

13= Rs E S(O f0) would be as desired. Otherwise by observation (b) at the

beginning of the proof t13 = Rs E S(P f0). Then t E S(P f0)n S(O f0) = S(P iF). But then tilE S(P f F) C S( 0) as desired. This completes the special case, hence the

proof.

Adding dependent choice

3.17. Forcing

For Theorem 3.2 forcing is defined as in 1.7 and 2.6 via: (a) Pll-*"c Ef"~(3g,f3)[PII-* " c = (f3,g)" l\ff3=pg] (b) P II-* c E J ~ (3/)[P II-* "c = /"] . For Theorem 3.1 the same definition works with l o replacing J.

3.18. Restriction Lemma

(a) Forcing automorphisms

133

As in 1.8 these are order preserving permutations of new function symbols. They

act as the identity on J (or l o). (b) Homogeneity Lemma (See 1.8(b). Replace f by j .) (c) Restriction Lemma. Let $ have parameters tn U U {J} U F.

p II- (/) ~ p I F II- $.

For Theorem 3.1 replace J by lo .

3.19. Models, support structures, and perceptions of forcing

In Theorem 3.2 .N' = U[J]. I,. is defined from J via Io = J n 2", I,. +1 = J n I~. J"·"' is defined in terms of 1 via I ... o =I,., 1 ... a+ l = J~:·<w.laiJ n I, and 1 ... "' = (U{J <aln.{J t n 1 for limit a. Set ],. = u a<K ln,a• The following are verified . (a) f E I,. maps K onto I,. _,, (b) ],. is an AD set of HAD functions with ranks < K on I,.. (c) For each a< K, n > 0, and f E I, there are g, · · · gk E / ,. _, such that f[ a] C

V{g, · · · gd. · The proof of (c) uses Lemma 3.10(b) and the observation that if Pis a condition

there is a Q ~ P such that for each n and a E A,.(Q) there is an f E F(Q) with a1 =a. (Simply take P and add f's to S(P) so that a1 =a for each a E A,.(P).)

If G is finite as usual let "V G be the class of sets definable in .N' from parameters in U U {J} U G. T( G, a) is the a th set in the "least definition" well ordering of "V G induced by some fixed canonical linear ordering on J. Every set of .N' is in some "V G.

The notion of satisfaction of a condition is defined exactly as in l.lO(a). The

satisfaction and truth lemmas follow as in 1.10(b) and l.lO(c). One typical consequence is the following.

(d) Characterization of "VG n J,.. If G C J,. and f E ],. then f E "V G ~ f E V G.

Proof. If f E VG then f = gf3, · · · f3k for suitable g E G and ordinals {3 1 • • • f3k· Therefore f E "V G. Conversely assume f~ V G but f = T( G, a) for some a. This is

forced by some P = P f G U {j} where G and f satisfy P. By 3.5(b) some ff3 1 • · • f3k E J,.- VG. Let 1T be a forcing automorphism fixing G and moving j. The

134 D. Pincus

proof of the compatibility lemma shows that P and 1rP are not only compatible but

have an extension R such that /f3~ · · · f3k ¢ R 7Tff3~ · · · f3k· Such an R forces T( G, a) to have two values, a contradiction.

The model for Theorem 3.1 is just U[ fo], a submodel of the model for Theorem

3.2. The above discussion can be relativized to this model with little change.

3.20 . Preservation of cardinals

(a) Pointed conditions Every condition P extends to a condition Q such that A.,(Q) = A.,(P) for the

highest n such that A .. (Q)I0 and A..,(Q) is a singleton for all m < n. Such a

condition is called pointed.

Proof. Use construction 3.1 O(b ).

(b) No new small sets If cp E }( maps A < K into U then cp E U .

Proof. Let cp E "V G. Each statement of the form cpa = {3 is decided by condition of

the form P r 6 where G satisfies P. Using (a) above it may be assumed that P is

pointed . The standard argument now applies. One builds an increasing sequence

Pa = Pa r 6 to decide all values of cp. At limit ordinals A.., (Pa) is constructed using

3.10(a).

(c) Preservation of cofinalities (b) above shows that cofi nalities ~ K are preserved. The collection of conditions

has cardinality K since K • = K. Thus cofinalities > K are preserved .

(d) Consequences (1) Every < K sized subset of I .. is included in V G for some finite G c J ...

Therefore if I X I < K and X C I .. Sp X exists. (2) Every g EJ .. is in "V{f} for every fE I., +l· g is defined as Sp(f[X]) for a

suitable X C K, I X I< K .

(3) From (2) above and 3.10(d) it follows that if G cJ .. , "VG n J =

U ... < .. J .. U VG. No g E I ... is in "VG for n < m since that would put J., c"VG.

3.21. Proof of DC<• in the model of Theorem 3.1.

Let R E U[fo] satisfy the hypotheses of DC' for A < K , R = T(f, {3) for some f E ]

0, {3 EOn. Notice that 3.10(b) permits the conclusion that every set is

supported by a singleton. Let p = p r I force all of the above. Using 3.10(b) and the fact that T(/, {3) is forced to satisfy the hypotheses of DC'

one can inductively build sequences Pa, aa, and y" for a ~ A which satisfy the

following. (a) Pa = Pa t j, a < 8 ~ P" ~ Ps, and Ao(Pa) is a singleton.

~dding dependent choice 135

(b) Whenever g is added to S(Pa) so that a8 is the element of Ao(Pa) the resulting Pa(g) forces "(T(g(as), Ys): o <a) is T(j, /3)-admissible." g(as) denotes that

t = g/31 · · · f3k such that a, = as. Carry the construction out to A and find a g such that g and f satisfy PA (g) (Truth

Lemma). (T(g(as), Ys : S <A) is an R -admissible A sequence.

3.22. Canonical supports

(a) Intersection lemma Let G~, G2 C f. be finite. If x E V' G1 n V' G2 then either (1) MCVG1Gd~ 0 and x E V'MCVG1G2.

, (2) MCVG1G2 = 0 and x E V'0 or V'G for some G Clm, m < n.

Proof. As in 1.11(c) assume x = T(G1, a)= T(G2, {3). Let P = P j 61 U 6 2 force the equality. P can be assumed to include inequivalent values for distinct elements

of H(P) . Let MCVG1G2 = {h1, ... , hd and fix some naming of each h, in terms of

the gi E G1 i.e. h, = gio/3 ; · · · 13 :<•>·

X= {g/31 · · · f3, E I. - I: g E G1 11 g/31 · · · f3, E S(P)}.

The function g/31 · · · f3, ~ g/31 · · · f3, can be defined in terms of Sp X since Sp X defines a well ordering of X and I X I< K. Therefore {x} is put into V'MCVG1G2 U SpX by the following definition .

{x}={T(G:,a):G; satisfies P, h,=g;,{3:···f3 :< •> for h,EMCVG1G 2 , and

g'/31 · · · f3, = g/31 · · · f3, for g' E G;, g/31 · · · f3, E I. _ ~, and g/31 · · · {31 E S(P)}.

The definition works because equality of terms between G; and G 2 must occur among values of MCV G1 G 2 or Sp X. Therefore such a G; U G2 satisfies P.

If MCV G1 G 2l0 then Sp X E V'MCV G1 G 2 by 3.20(d)(2). Otherwise x E VSp X. Of course if n = 0, X = 2 and x E V0.

(b) Canonical supports If x E JV (for Theorem 3.2) then x E V Gx for the unique Gx satisfying (1) Gx CJ. where n is least possible such that x E VG', G' CJ. or Gx = 0 if

X EV0. (2) If G' CJ. and x E VG' then Gx CVG'. (3) Gx is of minimal size satisfying (1) and (2) .

The same holds for Theorem 3.1. The only J. involved is ]0 •

3.23. Proof of Theorem 3.2.

3.20 shows that JV preserves the cofinalities of U and adds no sequences of length

< K to U. - C~ holds since (I" : n E w) does not have a choice function, essentially by 3.20(d)(3). 0 , and in fact KW, follows as in 1.12(b). It remains to prove C<K·

136 D. Pincus

Let 1 y 1 < K. A well ordering of Y will be defined from parameters in

u u {J} u { Y}. In fact since no cardinals are collapsed there will be a canonical map

from y to 1 Y 1 defined from these parameters. The method of well ordering Y is to

find a canonical G v such that Y C 'V G v and well order Y according to the

canonical well ordering of G v_ Actually G v can only be Gv according to 3.20(d)(2)

however it requires I Y I< K to prove that G v exist i.e. Gv works. To obtain G y let Xo = u yEY Gv. Let X,= Xo n ]., for the highest n with

Xo n ]., 'I 0. If Xo = 0, G v can be taken to be 0. There is a highest n with

Xo n ]., 'I 0 since Y is well orderable and every y E Y is supported by the support of a well ordering. I l., n V X,l < K by 3.5(c) so Sp(I., n V X,) is defined by

3.20(d)(I). G v is taken to be Sp(l., n V X,). If y E Y then Gv C I, for some m ::::; n. If m < n then G,, E'VGY and yE'VGv. If m=n then YGvCVX1 so

G,, E V Sp( l., n V X,) and again G"' hence y, is in 'V G v_

3.24. Proof of Theorem 3.1.

oc<K has already been proved in U[lo] and the proof of 0 immediately

relativizes downward from .N'. The proof of c<K in .N' actually relativizes to one of C<o .. in U[Jo) - The point is that if Y is well orderable in U[lo] it has a well ordering

supported by a G c lo. G v can be obtained as in Theorem 3.2 because every

Gv cVG. C<o .. fails in Theorem 3.2 because Gv needn't be included in VG. It can

be included in some lesser l.,..

4. A Fraenkei-Mostowski transfer theorem

Theorem 4. Let <P be a conjunction of any of the following 5 kinds of statements. If cp has a

Fraenkel-Mostowski model then </J has a ZF model. The 5 kinds of statements are:

I. Injectiuely boundable statements 2. c <", some i.L

3. DC<", some v 4. 0 5. Term -choice statements.

The use of ordinal parameters in the above statements presents the same sort of

problem as with the statement of Theorem 3.1 and Theorem 3.2. The exact

statement should be as follows.

Restatement Let .M be a Frankel-Mostowski model with well founded universe U satisfying

ZFE + G C H. Let </J be a conjunction as above which holds in .M. 1 here is a ZF model .N' preserving the cofina/ities of U such that cp holds in .N'.

Adding dependent choice 137

The reader should consult (17] for definitions and examples of injectively boundable and term-choice statements. In this paper I will not consider the general term-choice statement but only the typical special case , C<o ... [17] gives two theorems similar to Theorem 4. The first dealt with statements of the first 3 kinds . It also included C<0

". The second dealt with a weakening of injectively boundable statements as well as 0 and term choice statements. In [20] I added PI and HB to the second theorem. Theorem 4 represents a unification of the two theorems but it doesn't include all the statements of either one . It may be possible to use the present methods to add PI and HB to Theorem 4. It is not possible to do so with c<on.

A discussion of the uses of Theorem 4 appears in the conclusion to [17]. I mention here that Theorem 4 provides another way to add DC to the indepen­dence considered in Theorem 2.l(E). It also transfers C<.., in conjunction with arbitrary higher order statements about Dedekind finite cardinals. This is relevant to work of Ellentuck (see e.g. [2]). He considers the theory of the Dedekind cardinals under the assumption C< .... The results of [17] transfer higher order statements about Dedekind cardinals but not in conjunction with C<..,. Even Theorem 4 only works in the case that the class of Dedekind cardinals is a set

of M. At the end of this section I will sketch a possible way to add nc<• to the

independences of Section 2 without destroying Cw.

4.1. Fraenkel-Mostowski models

For the purposes of this paper Fraenkei-Mostowski models are of the type introduced by Mostowski [14], (also see [9] p. 47). More general models were introduced by Specker, (see [9] p. 46) . A theorem, to be proved elsewhere, states that every statement which is I2 in the power set operation has a Specker model if and only if it has a Mostowski model. This implies that the restriction is no loss of

generality . .st1 will denote the atoms of the Fraenkei-Mostowski model .;(;(. CfJ will be

automorphism group while 'lf is the ideal of supports, i.e. well orderable subsets of .stJ V is. the support relation of .;(;( and T( W, a) is the a th element of the well ordering of V H induced by the well o rdering W of HE 'lf. Up to now in this paper models have all had the property that a support has a canonical well ordering. This is not true in general , which is why one must talk of T( W, a) rather than T(H, a).

For x E.;(;( and HE :Je [x ]H denotes the orbit of x under those elements of CfJ

which fix H pointwise. [x ]H is definable in .;(;( as the unique minimal (under inclusion) element of V H which contains x .

.;(;( can be coded in U in the sense that the re is a class .;(;( * C U such that .;(;( * is E

isomorphic to .;(;( . .;(;( * is transitive except that 0At' and individuals of .;(;( * are nonempty sets of U . .;(;(* will be identified with .;(;( in the sequel.

138 D. Pincus

4.2. Constants

.;U will be mounted on top of the base set Io (now to be called I) of the model of Theorem 3.1. The regular cardinal K chosen will depend on the statement cp to be transferred. For the moment let K be arbitrary. Recall that the GCH in U implies that K < K = K. The new constants are as follows.

(a) The new function symbols for Theorem 3.1 which have orders a < K (corre­sponding to (0, a)). (b) A constant J, corresponding to l o of Theorem 3.1. (c) Constants for the members of At.

The new function symbols and 1 are placed as. in Theorem 3.1. The potential members of p E st1. are the new function symbols of order 0. The potential members of x EAt are its At-members.

4.3. Conditions

Conditions are pairs P = (P*, tf;p) where P* is a condition of Theorem 3.1 and t/JP maps H(P *) into st1. such that Range tf;p E 'X. P..;; Q if P*..;; Q* and the function u which illustrates P*..;; Q * (see 3.14) satisfies t/Jou = t/fp. Equivalent conditions and restrictions to F are defined as in Section 3. The compatibility lemma is stated and proved as in 3.16.

4.4 . Forcing

Strong forcing of membership in new function symbols and J is as in Section 3. If

p E .stJ.,

P If- * " c E p" ~ (3/, h )[P If-* "c ·= f" 11 f=p h 11 tf;ph = p].

For other r E AlP If-* "c E r" ~ (3q EAt n r)[P If-* "c = q"] .

The group of forcing automorphisms is f!F X W where f!F is the group of Theorem 3.1. f!F acts as the identity on J and At while W acts as the identity on J and new function symbols. Homogeneity and restriction follow as in Section 3.

4.5. The model and support structure

In .N' J is once again an AD set of HAD functions on I. st1. is a disjoint partitioning of I and At is (from the viewpoint of the full generic extension) isomorphically embedded above st1. and is transitive except for members of .stJ.. The isomorphism doesn't exist in .N' of course.

A normal support is a pair G, H where G C J is finite and H is a support of .M such that H ::J {p E st1.: (3/ E In VG)[f E p ]}. If WE .M well orders H, W induces a well orderin~ of V' G, H. T( G, W, a) is the a th element of this well ordering. Every x E .N' is in some VG, H . V'G, H contains G, the .M-well orderings of H, and

. is closed under definability from parameters in U U {J} U (''V H).«.

Adding dependent choice 139

Satisfaction of conditions is slightly tricky to define in X. In .All the map taking (;]0

. to [r] is defined since pis the code of pin .All* (see 4.1.). The proper thing to say is

that G and H satisfy P for 6, H if G satisfies P * and [ tPI' ]0 goes under the above map to the 0-class in .All of the membership function of .sli (the function taking f E I to the unique member of .sli containing it).

As in Section 3 f E VG, H n I~ f E VG. Cofinalities are preserved from U to .N and no new functions from A to U are in .N where A < K .

- . The support intersection lemma takes the following form. If G., H and

G2H are normal supports then VG,, H n VG2, H = VMCVG,Gz, H. If x E VG,, H, n VG2, H2 then x E VMCVG,Gz, H, U H 2. Thus the canonical sup­port lemma applies to the G-part in the sense that for every x E }( there is a

canonical Gx such that for some (not necessarily canonical) H, x E 'ilGx, H. The proof of Theorem 3.1 also yields that for every well orderable YEN there is some

H such that Y C V Gv, H.

4.6. The transfer of 2:, statements in power set

This is essentially Jech-Sochor. See [9] . Let VH = U a. Hnormal VG, H.

(a) Lemma If p E .All, 01 y C ([p ]H y«, and y E V H then y = ([p ]H)".«. Thus ([p ]H)"« is also

definable in}( as the unique minimal member of V H containing p. The .All superscript is henceforth dropped.

Proof. Use forcing automorphisms 7T E ~ C fJi x ~ such that 7T fixes H pointwise and maps a member of y arbitrarily in [p ]H.

(b) Lemma If Y C .All and, relative to }(, Y does not map onto K then Y E .All.

Proof. Let Y C p .All. By (a) above Y is a well ordered union of sets of the form [g ]H for some fixed H. There are < K cells to this union so, since there are no new small

sets of ordinals, Y E .All.

(c) Lemma · If x E .All then x maps onto K in }( exactly when it does in .All.

Proof. See (a) above and the preservation of cardinals.

, (d)Transfer of 2:, statements in power set . Such a statement is 3x<P(x) where <P is !o in power set. It follows that for some rank over x, R., (x), <P(x)~(<P(x)ta< • l. Let p be an instance of <P(x) in .All and let K

be large enough so that (R, (p ))".« does not map onto K . By (b) and (c) above

(R .. (p)}.« = Ra(P) and p remains an instance of cP(x) in X.

140 D. Pincus

4.7. The transfer of injectively boundable statements.

The a rgument of [17] Section 1 needs two facts. The first is the transfer of 1;1

statements in power set, as shown above. The second is the following lemma.

(a) Lemma For given At and A there is a bound b ;:::: A definable in At such that whenever the K

of .N exceeds b the following holds in .N. "If A does not map 1:1 into x then x has the cardinal of an element of At."

Proof. Whenever A< K and A does not map 1:1 into x it is claimed that Gy E VG. for every y E x. Let W well order some H such that both G., H and Gy, H are normal and X= T(G., w, a) while y = T(Gy, w, (3). Suppose p = p r Gy u G. forces T(Gy, W,a) and Gy~VG •. Without loss of generality assume that A(P*) is a singleton of rank 8 ;:::: A.

Let (P~, y < 8) be a sequence o!_!epresentatives for p r G. such that the identity gives p r 6. :,;:; py but H(P~,) n 'H (Pn) = H(P r G.) for Yt-1 'Y2· Let p+ be the condition formed by letting S (P+) = U ~ < s S (P~ ), introducing no = P+ equivalences other than those of Py, letting A (P+) ={a} where ay is the element of A (P~), and letting t/J p + = U y<s t/JPy· Form p ++ by introducing j to S (P+) such that a1 = a.

Use the truth lemma to find an f such that f and G. satisfy p + +. Set

G~ = {fyf3t · · · f3t: (3g E Gy){3c)[A(P*) = {c} A cf3t · · · {3y = a 11 ]}.

The G~ all satisfy P along with G •. Furthermore if 'Yt I Y2 then MCVGs,G.s, c G. (see 3.5(b).). Let Y~ = T(G~, H, (3). Y~ EX and GY> = G~ since Gy satisfies P. The map y- yy sends 8;:::: A into x. A does not map 1:1 into x so there are y 1 -l y 2 such

that yy, = Yn· y~, E V( Gy, H) n V( Gn, H)= VMCV Gy, Gn, H. This contradicts G y ~VG. and verifies the claim made at the beginning of the proof. >,

From the claim every y Ex has the form T(G., W, (3) for Wand {3 depending on y. Let {3y be the least possible (3 as above and let 'Wy be the set of all W's such that y = T( G., W y, {3y ). Pick the lower bound b for K so that the range of the 1:1 map y- ('Wy, {3y) is in At. This can be done by guaranteeing that the set of sets of well orderings of subsets of s1 is absolute from At to .N. In that case there are <A (3 ' s such that ('W, (3) is in the range of the map for any fixed 'W. The range can't map onto K so 4.6(b) puts the range in At. The set of sets of well orderings of subsets of s1 can be made absolute by 4.6(b) and (c).

4.8. The transfer of C< ~<

For given At and A there is a bound b ;:::: A definable in At such that whenever the K of .N' exceeds b (CA )M - CA holds in .N'.

Proof. The bound on K is such that the set of well orderings of sets of well

Adding dependent choice 141

orderings of subsets of .szt is absolute from .M to .N. Such a bound can be defined by 4.6(b) and (c). Assume that K exceeds the bound and C' holds in .M for A < K.

Let (Xa : a < A) be a sequence of non empty sets in .N. Let W, be the set of well orderings of supports, H, such that V H contains a member of Xa. Recall that V H = UoHnormaJ VG, H. By the absoluteness and CAin .M there is a choice sequence from ("Wa :a <A) in .M. Using this sequence one can define a well ordering W of a support H such that V H contains a member of Xa for each a.

One can now use either the proof of DC<K in 3.21 or the argument producing the

(G.r: 'Y < 5) in 4.7 to obtain a G such that VG, H contains a member of Xa for each a. One can then choose from Xa the least (according to T(G, W, {3)) member of

Xa nVG,H.

4.9. The transfer of DC<•

For given .M and A there is a lower bound b ~A definable in .M such that whenever the K of .N exceeds b, (DCA)-« ~DCA holds in .N.

Proof. The lower bound on K is such that the set of relations on well orderings of subsets of .szt is absolute from .;{;t to .N. The proof of (DCA ).M ~DCA will be omitted.

It combines the proof of DCA in 3.21 with the transfer of DC' of (17].

4.10. The transfer of 0

There is a lower bound b definable in .M such that whenever the K of .N exceeds b, o-« ~ 0 holds in .N.

Proof. The bound is such that the set of sets of well orderings of subsets of .szt is

absolute from .M to .N. If 0 is true in .M there is a linear orderi!)g of this set. For x E .N let a, be least such that for some W, x = T( G., W, a.). Let W, be the

set of those W. x ~ (G., W., a , ) maps the universe 1:1 into an orderable class.

4.11. The transfer of term choice statements

. The typical special case c <On is considered. As before the statement is that there

is a bound b definable in .M such that if K exceeds b then (C<on )--« ~ C <on holds in .N .

. Proof. Choose the bound so that the set of sets of sets of well orderings of subsets of .9'1 is absolute from .M to .N. Assume this is true and c <On holds in .M.

Let X be a well orderable set in .N. Every x E X has the form T( G x, W , a) by a

remark of 4.5. Define a , and W, as in 4.10 using G x instead of G, and consider the 1:1 map x ~ (W., a.). The set of W. occurring in the range of this map is well orderable so the absoluteness and C <on in .M chooses an element W. Choose from X that x such that W. = W and a. is least.

142 D. Pincus

4.12. Proof of Theorem 4.1.

In view of 4.6, ... , 4.11 it suffices, for th e given cJ> and .M to take K greater than a

finite number of lower bounds.

4.13. An idea for adding DC<" and C<on to Theorem 2.1(A), (B) and (C)

Form the Fraenkel-Mostowski model using a K -saturated structure. Note that this model will not satisfy the appropriate positive statement (e.g. C<w). Form the. ZF model as in this section . Then add an AD set of HAD functions on the supports of the Fraenkel-Mostowski model.

This idea seems to work for Theorem 2 C, D and E but not for A and B.

5. Adding countable choice without dependent choice

Theorem 5 cw can be added to the independence of DC from 0 1\ c<On· Subject to the usual reservations about ordinal parameters the following

generalization is possible .

Restatement and generalization

Let K and A be regular cardinals where K < = K. DCA . is independent of 0 1\ DC<A 1\ c << 1\ c <On•

Theorem 5 will be considered as stated. The generalization to K and A is straightforward. The idea is to add the ordering theorem, by means of HAD functions, to 1 en sen's original independence proof of DC from cw. The reader can consult [9) for an ex position of the independence of DC from C".

5 .1 . Constants

The model is essentially that of Theorem 3.2 except that the elements of I are also given the tree and support structure of Jensen's proof. The new constants are those of Theorem 3.2 together with a constant ~ whose possible members are ordered pairs from I . ~ will become the tree structure on I.

5.2. Conditions

A condition is a pair (P, < )where P is a condition of Theorem 3.2 and < is a tree structure on H(P) with the following additional properties. (a) There is a single base node. (b) Every node has finite height.

Adding dependent choice 143

(c) There are no w-paths through (H(P), < ). (d) If a E V(A (P)) has order 1, r <sin H(P) , and s =an for some n then p =am for some m.

5.3. The ordering of conditions

, (P, <) ~ (Q,<') if P ~ Q in the sense of Theorem 3.2 and the map a giving the embedding is both height preserving and order preserving from (H(P), <) to (H(Q),<').

5.4. Forcing

The forcing of membership in the constants of Section 3 is as there, (f, g) E ~ is strongly forced by (P, <) if for some r, s E H(P), f==pr, g == ps, and r < s.

5.5. Support intersections and the ordering theorem

In the model resulting from 5.1-5.4 we call a finite G C J normal when I is closed under ~ predecessors. The only concern is for f E I since 5.2(d) guarantees such closure for values of higher f's. The claim is now that if G. and G z are normal supports then so is MCVG.G2 and furthermore VG. n VG2 = VMCVG.G2.

MCV G 1 G 2 is immediately seen to be normal. Assume x E V G. n V Gz. As in the proof of 3.22 equality is forced between G. and G2 terms denoting x by a p = p t {a. u G2} which distinguishes between distinct elements of H(P). It suffices as usual to show that if G; satisfies (P f G., < f H(P f G.)) over MCVG1G 2 then G; U G 2 satisfies (P, < ). G; U G2 satisfies P by 3.22. It satisfies < because a E Ia,- I Mc va,a2 and b E I a, - I Mcva,a 2 behave the same way under ~ as the corresponding a' E Ia; - I Mcva,a2 and b. In fact both are ~-incomparable.

Otherwise, since ~ is a tree structure and all the supports are normal , the lesser of a and b or a' and b is in MCVG.Gz.

Using the support intersection lemma one proceeds as in 3.22(b) to a canonical normal support for the given X; i.e., first minimize a such that (3G c ull<? Jil). [x E VG], notice that there is a canonical minimal G n l a, and proceed to lower orders, terminating when order 0 is reached. The ordering theorem is now immediately seen to hold in the model.

5.6. C"' and -,DC in the model

The w-tree (I,~) constitutes a failure of DC since easy symmetry arguments coupled with clause 5.2(c) in the definition of condition imply that no infinite path through the tree is supported. The proof of C"' proceeds as follows.

Let (X": nEw) be a sequence of nonempty sets. As in 3.20(a) it can be assumed that the sequence is supported by the singleton g. Let P" = P" f {j", g} and <" be

144 D. Pincus

such that g satisfies (P .. ,< .. ) and (P,, < .. ) forces both g E Vf and Vj .. n X" -10. Without Joss of generality the j .. can all be assumed to have the single order a < w 1•

As in 4.7(a) one can put together a master condition P = P f {fi, g} satisfying the

following: (a) 1i has order a + 1. (b) When h and g satisfy P then h .. and g satisfy P ... (c) When h and g satisfy P and n 'I m then Ih. n hm = [8 •

Let < be defined to extend the isomorphic copy of <" induced by letting j. correspond to li". Two elements of H(P) not in the image of the same Vj" will be <-incomparable. (P, <)is such that if h, g satisfy it then Vh contains a member of

each X". Thus the X" have a choice function.

Glossary of choice properties

If {XyLEv is an indexed family of sets a choice function is a function tf; on Y such that 1/Jy E Xy whenever Xy '10. A choice function can also be called an element selector since it selects an element whenever possible . In a similar spirit one could speak of a subset selector, a a -element subset selector, etc . .M and J{ denote

cardinal numbers. c-; ~"Every .M-sized family of .N -sized sets has a choice function." One can substitute < .M for .M or < .N for .N with obvious interpretations. <On should be read "well orderable." (On denotes the class of ordinal numbers.)

c" ~vxc"!::x Cx ~v.Mc-; AC~V.MC,~1

If R is a binary relation on X, aEOn, and sEX" s is R admissible if

(V{3 < a)[s f {3Rs{3]. DC~ "If for every a < K on R admissible a-sequence extends to an R admissible a + 1-sequence then there is an R admissible K -sequence."

oc<· ~<VA < K)DC' oc~ocw ~(Vx E X)(3y E X)[xRy]-(3s E X w)(Vn E w)[snRs(n + 1)].

A Dedekind finite set is an infinite set with no w-sized subset. Z(.M) ~"Every family of sets not smaller than .M has an .M-sized subset selector." Z( < .M): ~"Every family of nonempty sets has a < .M-sized nonempty subset

selector." SP ~ "Every family of nonempty nonsingleton sets has a proper subset selector." KW~"Every set can be mapped 1:1 into the power set of an ordinal."

0 ~"Every set can be linearly ordered." PI~ "Every Boolean algebra has a prime ideal." HB ~ "Every Boolean algebra has a finitely additive nonnegative measure which is

positive on 1." GCH ~ (V.M~ w )(V.N')[.M~ .N v .N~ 2.41] E ~"The universe has a well ordering ."

Adding dependent choice 145

Added in proof. I have recently proved PI in the model of Theorem 3.1. The methods permit one to add PI and HB to the Fraenkei-Mostowski transferable Statements of Theorem 4.

References

(1) P.J. Cohen , Set Theory and the Continuum Hypothesis (Benjamin, 1966). (2) E. Ellen tuck, The universal properties of Dedekind finite cardinals, Ann. Math. 82 (1965) 225-248. (3) U. Feigner, Uber das Ordnungstheorem, Z. Math . Logik Grund. Math. 17 (1971) 257-272 . (4) U. Feigner, Independence of the prime ideal theorem from the order extension theorem, to appear. (5) R. Gauntt, The axiom of choice for finite sets - A solution to a problem of Mostowski,

unpublished. (6) P. Howard, Limitations on the Fraenkel-Mostowski method of independence proofs, JSL 38 (1973)

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